MUSINGS OF A PHILOGAMER 007 Cursed? On the Gambler s Fallacy, Confirmation Bias, and the Case of Mini War Gaming s Quirk Matthew Gray Army Gaming 05 August 2017 1. Introduction Consider, for a moment, the following scenario. You and a friend have decided to hammer out a few rounds of your favorite table-top wargame. Although neither of you has any clear advantage at the outset, within a few minutes you begin to feel that the victory has started to elude your grasp. Roll after roll things just do not seem to go your way as the dice reveal string upon string of frustrating numbers. It looks as if the dice gods have turned against you as you sit helplessly and watch your army gradually crumble. In short course the game comes to an end with you suffering a massive defeat. Given this experience you determine the most reasonable conclusion is, simply, that you are cursed. Long-time gamers as well as newcomers will most certainly recognize themselves in this story. We have all had the experience of entering a dice-casting drought where our rolling hand seems to be as cold as Dwarven steel left out in the Norse winter. Of course, few (if any) of us thinks that we are literally cursed. That is, the language of curses is merely employed in a metaphorical manner. Even so, we oftentimes do end up with some vague belief that there is some pattern at work in our misfortune that tips the scales of fate against us. Beyond this point, however, few of us ever venture. In what follows, I would like to explore the phenomenon of the curse of the dice rolls. For those of you who find the idea of a curse appealing or who enjoy the mystique of the possibility 1
of unknown forces influencing your dice rolls, my argument will most likely be rather disappointing. In short, I claim that although strings of bad rolls (e.g., a long string of 1s) certainly occur, these stings are rarely, if ever, a sign of any deviation from statistical averages. Instead, the idea of a curse can be traced back in part to two different forms of fallacious thinking, namely the Gambler s Fallacy and Confirmation Bias. To begin, we sketch out the Gambler s Fallacy and Confirmation Bias in turn after which we then take up the case of the Mini War Gaming employee Quirk and the apparent curse of bad rolls (i.e., 1s) under which he suffers. In relation to this case we will carry out our philosophical and statistical analysis as well as put forward several reflections on probability and human psychology. 2. The Gambler s Fallacy Let us begin with the Gambler s Fallacy. The Gambler s Fallacy, what is also known as the Monte Carlo Fallacy or the Fallacy of the Maturity of Chances, is an incorrect manner of reasoning by which one assumes that independently random events are in fact influenced by surrounding and past events. A good example of this is how some people understand the roulette wheel at a casino to function. If the wheel is constructed in a fair manner, each spin operates as an isolated random event in which a ball, after being whirled around the wheel, lands on one of 37 or 38 colored and numbered pockets. 1 Despite this, it is common for both operators and gamblers to believe that a string of occurrences, for example, 12 rolls of black, entails that the roll of red will be more likely the next time. This, however, is incorrect. As Stuart A. Vyse nicely puts it, Each spin of the wheel is an independent event and has no effect on subsequent spins. Even if twenty-five blacks were rolled in a row, the probability of a red on the next spin would be unchanged. 2 To use an even simpler example, the chances of flipping heads on a coin are always 50/50 irrespective of the result of preceding flips. Indeed, this fallacious way of thinking has had some deleterious consequences such as in the case of the patrons of the Monte Carlo Casino from which the fallacy gains its name. On August 18, 1913 gamblers within the casino received word that the roulette table was rolling a long streak of black and consequently they rushed the table in order to bet on the inevitable red. Red, however, was long coming not before 26 consecutive spins of black. And in the long road to 26, betters, doubling-down in their belief that red had to be more likely on the next spin, lost millions of francs to the casino. The gambler s fallacy is perhaps a far more common occurrence in our lives than we would care to admit. It shows up in our beliefs about sports events such as the cold streak in a batter or the hot hand in basketball, in the pride we take in our belief that we are able to predict 1 The number varies depending upon whether one is referencing the European or American wheels. 2 Stuart A. Vyse, Believing in Magic: The Psychology of Superstition (Oxford: Oxford University Press, 1997), p. 98. 2
future events (that are in fact random), in our pointing to underlying reasons for why one person was infected with a sickness and another was not, and so on and so forth. And part of the reason we fall prey to this fallacy time and again is directly tied to our humanity. As human beings we seek narrative unity to what happens around us perhaps as a rebellion against the uncomfortableness of randomness. Moreover, we remember what happens in the past and we are oftentimes unable to bracket these memories when we are faced with a new independent event. Put differently, we don t think as purely logical robots, but rather we think as beings with memories who are embedded within a narrative arc. Thus arises the Gambler s Fallacy. 3. Confirmation Bias A second form of fallacious thinking that we may touch upon is that of Confirmation Bias. In short, Confirmation Bias represents the irrational tendency to interpret or remember information in such a way that confirms one s already-held beliefs and affirms one s own hypotheses. 3 Consider for example the case of astrology and horoscopes. 4 Horoscope.com has listed the following for Pisces (my own sign) for today s date, August 5, 2017: You may have had a tendency to go about your daily routine lethargically lately, Pisces. But today the alarm clock wakes you up. You may understand that your help is urgently needed and there s no time to waste. You can expect to pour a great deal of energy into a single, well-defined goal. If you usually wander from project to project, this will change for you. 5 Now, a Pisces who believes in the power of astrology will likely find this message to be insightful. Why is this the case? Aside from the rather generic language of the horoscope which allows it to be applicable to almost any reader, the believer in astronomy interprets the message in a particular manner by which he lays greater emphasis on aspects of his life that conform to the description while conveniently overlooking aspects that contradict it. In other words, the faithful Pisces has a bias to believe in the truth of the message and therefore he ultimately ends up with evidence that confirms the belief he already holds. In many areas of life Confirmation Bias represents a dangerous breakdown of proper thinking: for example, when scientists are led to draw conclusions that affirm their scientific hypothesis but conclusions that are in fact false; or when we draw prejudiced conclusions about entire groups of people being a certain way because we notice and remember a few affirmative cases while missing altogether the overwhelming number of counterexamples with which we 3 See Frederic P. Miller, Agnes F. Vandome, and John McBrewster, Confirmation Bias (Saarbrücken, Germany: VDM Publishing, 2009). 4 See Vyse, Believing in Magic, pp. 120-1. 5 https://www.horoscope.com/us/horoscopes/general/horoscope-general-daily-today.aspx?sign=12. Retrieved August 5, 2017. 3
are met on a daily basis. As such, it is crucially important that we remain vigilant in questioning our own biases and retaining a healthy skepticism about the conclusions we draw that, on the surface, appear to be so obvious. 4. The Case of Mini War Gaming s Quirk: Statistical Analysis But what, more specifically, do the Gambler s Fallacy and Confirmation Bias have to do with the notion of being cursed in wargaming? In moving towards answering this question, we may now take up the case of Mini War Gaming s Quirk through the lens of statistical analysis. As we mentioned in the introduction above, the story about the curse of Quirk has long been floating around Mini War Gaming headquarters, whether in numerous batreps or in various Sit and Talks. As we would be led to believe, Quirk has been doomed to roll 1s. When we carry out statistical analysis of a sampling of the games in question where Quirk has been identified as cursed, the data nevertheless reveals something far different. Consider three examples. First, in the Return to Fayoom: Mission 4A 6 batrep we notice that out of 706 rolls Quirk in fact rolled 1s a mere 109 times. This is 15.4% of the time, which is 1.3% below the statistical average of rolling a 1 on a D6. 7 In contrast, Steve, out of 507 rolls, ended up with a total of 96 cases of 1s, which is 2.2% over the statistical average. Moreover, when we consider 5s, which generally is a good number, Quirk was far over the average by 3.1% and Steve was in fact far under the average by 2.1%. From this data, we see that Steve would instead be a better candidate for the one who is cursed. Second, in Old World Wars Episode 223 we have a similar result. Here Quirk rolls almost entirely in-line with the statistical averages and even ends up at 0.5% below the average for rolls of 1s. Steve, on the other hand, shows greater deviation: not only is he massively under the average with respect to his rolls of 4s, but he is also 1.5% over the average with respect to the roll of 1s. Again, in this batrep Steve would appear to be the one who suffers from the curse. Finally, in War of the Realms Episode 145, we see that although Quirk is over the average with respect to the roll of 1s, he is nevertheless over by a mere 0.6%. Such a deviation can hardly be construed as anything like a curse. In conclusion, then, as the statistical data points out, Quirk is in fact snuggly aligned with the laws of probability. Or to phrase this another way, we simply have no evidence of Quirk suffering from anything like a pattern of bad luck. Briefly stated: no curse. 6 This is an important batrep precisely because Steve, in one of his Sit and Talks identifies it as one of the examples where the curse of Quirk is clearly manifested. 7 These percentages have been rounded. 4
5. Concluding Remarks: The Fallacies and the Curse With our conclusion concerning the curse of Quirk, we have now come full circle. What exactly can account for the discrepancy between the statistical data and the belief at Mini War Gaming regarding Quirk s misfortune? In short, we can point to the phenomena of the Gambler s Fallacy and Confirmation Bias that we have sketched out above. With respect to the latter, it is not difficult to imagine some form of this fallacious thinking at play at Mini War Gaming. Perhaps at some crucial point in a game Quirk rolled an untimely 1, which in that particular situation proved to be rather damning. Such an occurrence likely became the center of stories and jokes for the rest of the day, the week, and the month, which in turn embedded the occurrence into the memory of the content producers and began to create the narrative of Quirk as unlucky. In subsequent games, then, Quirk and Quirk s opponents suddenly began to notice 1s everywhere while inadvertently ignoring the statistical weight of these rolls in relation to the other rolls. The bias had now taken root, and each 1 rolled only served to confirm and to solidify the belief that Quirk was cursed. With respect to the former, once the narrative of Quirk as cursed took hold, it would then become easy for the Mini War Gaming staff to slide into a form of the Gambler s Fallacy (albeit an inverse form) in which they began more and more to believe that the string of 1s they believe they have observed would, due to Quirk s being cursed, entail that more 1s would certainly be on their way. And in this way, the curse of Quirk becomes legend, all the while lacking any basis in statistical evidence. It is important, however, to note that even though Quirk s rolls generally fell in-line with the laws of probability, such rolls nevertheless did occur at different times and in different situations within the games played situations that were more or less beneficial to Quirk. In this respect, it could be possible that the curse felt by Quirk was a result not of rolling 1s, but of rolling a disproportionate number of bad rolls in particular situations (e.g., rolling low when attacking but high when taking leadership tests). Fleshing out this possibility would require further analysis that runs far beyond the scope of this essay. My hypothesis, however, is that even in these cases there is likely not adequate deviation from the statistical norm to claim that Quirk is any worse off with respect to probability than any of his opponents. In the end, perhaps we the gamers see Quirk as cursed because we are not always thinking logically and rationally. Rather, we are instead trying to construct a narrative that fits into the arc of our gaming lives and that carries the semblance of coherence with our gaming memories. But perhaps this is precisely what we are after when we play these games: we are looking to suspend reality and to believe in magic. And perhaps such suspension and belief, if kept to the wargaming table and to our gaming halls, is harmless enough to be entertained and embraced. 5
Appendix: Statistical Breakdown of Dice Rolls from Three Games Return to Fayoom: Mission 4A Quirk +/- Steve +/- Total Rolls 706 507 1 109/15.4% -1.3% 96/18.9% +2.2% 2 114/16.1% -0.6% 82/16.2% -0.5% 3 124/17.6% +0.9% 90/17.8% +1.1% 4 104/14.7% -2.0% 85/16.8% +0.1% 5 140/19.8% +3.1% 74/14.6% -2.1% 6 115/16.3% -0.4% 79/15.6% -1.1% Old World Wars: Episode 223 Quirk +/- Steve +/- Total Rolls 452 593 1 73/16.2% -0.5% 108/18.2% +1.5% 2 72/15.9% -0.8% 110/18.5% +1.8% 3 80/17.7% +1.0% 105/17.7% +1.0% 4 79/17.5% +0.8% 78/13.2% -3.5% 5 70/15.5% -1.2% 92/15.5% -1.2% 6 78/17.3% +0.6% 100/16.9% +0.2% War of the Realms: Episode 145 Quirk +/- Mitch +/- Total Rolls 816 547 1 141/17.3% +0.6% 82/15.0% -1.7% 2 150/18.4% +1.7% 98/17.9% +1.2% 3 125/15.3% -1.4% 100/18.3% +1.6% 4 144/17.6% +0.9% 105/19.2% +2.5% 5 123/15.1% -1.6% 82/15.0% -1.7% 6 131/16.1% -0.6% 80/14.6% -2.1% 6